Information of spin space group $P_{I}\ce{^{1}{\text{-}4}}\ce{^{\infty m}{1}}$ ( nSSG:81.82.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
b, -a, -c | x, y, z	{ 1 ‖ -4+001 | 0 }
-b, a, -c | x, y, z	{ 1 ‖ -4-001 | 0 }
a+1/2, b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 2001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -4+001 | 1/2 1/2 1/2 }
-b+1/2, a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -4-001 | 1/2 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
b, -a, -c | -x, y, z	{ m ‖ -4+001 | 0 }
-b, a, -c | -x, y, z	{ m ‖ -4-001 | 0 }
a+1/2, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -4+001 | 1/2 1/2 1/2 }
-b+1/2, a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -4-001 | 1/2 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
2b	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
2c	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(0,1/2,1/4 | 0,0,0)
2d	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(0,1/2,3/4 | 0,0,0)
4e	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
4f	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
8g	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
A:(1/2,1/2,1/2)	$\ce{^{1}{-4}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{-4}}\ce{^{\infty /mm}{1}} $
M:(1/2,1/2,0)	$\ce{^{1}{-4}}\ce{^{\infty /mm}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{-4}}\ce{^{\infty /mm}{1}} $
R:(0,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
X:(0,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Λ:(0,0,w)	$\ce{^{m}{-4}}\ce{^{\infty 2}{1}} $
V:(1/2,1/2,w)	$\ce{^{m}{-4}}\ce{^{\infty 2}{1}} $
D:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Δ:(0,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
S:(u,u,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,u,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
T:(u,1/2,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
U:(0,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
W:(0,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Y:(u,1/2,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(0,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $
C:(u,u,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $
F:(u,1/2,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

2a

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_{1}^{S}(2) $
Γ:(0,0,0)	$Γ_{1}^{S}(2) $
M:(1/2,1/2,0)	$M_{1}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2) $
R:(0,1/2,1/2)	$R_{1}^{S}(2) $
X:(0,1/2,0)	$X_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2) $
V:(1/2,1/2,w)	$V_{1}^{S}(2) $
D:(u,v,0)	$D_{1}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2) $
E:(u,v,1/2)	$E_{1}^{S}(2) $
S:(u,u,1/2)	$S_{1}^{S}(2) $
Σ:(u,u,0)	$Σ_{1}^{S}(2) $
T:(u,1/2,1/2)	$T_{1}^{S}(2) $
U:(0,v,1/2)	$U_{1}^{S}(2) $
W:(0,1/2,w)	$W_{1}^{S}(2) $
Y:(u,1/2,0)	$Y_{1}^{S}(2) $
B:(0,v,w)	$B_{1}^{S}(2) $
C:(u,u,w)	$C_{1}^{S}(2) $
F:(u,1/2,w)	$F_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

2b

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_{2}^{S}(2) $
Γ:(0,0,0)	$Γ_{1}^{S}(2) $
M:(1/2,1/2,0)	$M_{1}^{S}(2) $
Z:(0,0,1/2)	$Z_{2}^{S}(2) $
R:(0,1/2,1/2)	$R_{1}^{S}(2) $
X:(0,1/2,0)	$X_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2) $
V:(1/2,1/2,w)	$V_{1}^{S}(2) $
D:(u,v,0)	$D_{1}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2) $
E:(u,v,1/2)	$E_{1}^{S}(2) $
S:(u,u,1/2)	$S_{1}^{S}(2) $
Σ:(u,u,0)	$Σ_{1}^{S}(2) $
T:(u,1/2,1/2)	$T_{1}^{S}(2) $
U:(0,v,1/2)	$U_{1}^{S}(2) $
W:(0,1/2,w)	$W_{1}^{S}(2) $
Y:(u,1/2,0)	$Y_{1}^{S}(2) $
B:(0,v,w)	$B_{1}^{S}(2) $
C:(u,u,w)	$C_{1}^{S}(2) $
F:(u,1/2,w)	$F_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

4e

k-vector	A↑G(4)
A:(1/2,1/2,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
M:(1/2,1/2,0)	$M_{1}^{S}(2)⊕M_{2}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
R:(0,1/2,1/2)	$2R_{1}^{S}(2) $
X:(0,1/2,0)	$2X_{1}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2) $
V:(1/2,1/2,w)	$2V_{1}^{S}(2) $
D:(u,v,0)	$2D_{1}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2) $
E:(u,v,1/2)	$2E_{1}^{S}(2) $
S:(u,u,1/2)	$2S_{1}^{S}(2) $
Σ:(u,u,0)	$2Σ_{1}^{S}(2) $
T:(u,1/2,1/2)	$2T_{1}^{S}(2) $
U:(0,v,1/2)	$2U_{1}^{S}(2) $
W:(0,1/2,w)	$2W_{1}^{S}(2) $
Y:(u,1/2,0)	$2Y_{1}^{S}(2) $
B:(0,v,w)	$2B_{1}^{S}(2) $
C:(u,u,w)	$2C_{1}^{S}(2) $
F:(u,1/2,w)	$2F_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

4f

k-vector	A↑G(4)
A:(1/2,1/2,1/2)	$A_{3}^{S}A_{4}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
M:(1/2,1/2,0)	$M_{3}^{S}M_{4}^{S}(4) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
R:(0,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
X:(0,1/2,0)	$X_{1}^{S}(2)⊕X_{2}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2) $
V:(1/2,1/2,w)	$V_{2}^{S}V_{2}^{S}(4) $
D:(u,v,0)	$2D_{1}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2) $
E:(u,v,1/2)	$2E_{1}^{S}(2) $
S:(u,u,1/2)	$2S_{1}^{S}(2) $
Σ:(u,u,0)	$2Σ_{1}^{S}(2) $
T:(u,1/2,1/2)	$2T_{1}^{S}(2) $
U:(0,v,1/2)	$2U_{1}^{S}(2) $
W:(0,1/2,w)	$W_{1}^{S}(2)⊕W_{2}^{S}(2) $
Y:(u,1/2,0)	$2Y_{1}^{S}(2) $
B:(0,v,w)	$2B_{1}^{S}(2) $
C:(u,u,w)	$2C_{1}^{S}(2) $
F:(u,1/2,w)	$2F_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8g

k-vector	A↑G(8)
A:(1/2,1/2,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2)⊕A_{3}^{S}A_{4}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}Γ_{4}^{S}(4) $
M:(1/2,1/2,0)	$M_{1}^{S}(2)⊕M_{2}^{S}(2)⊕M_{3}^{S}M_{4}^{S}(4) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2)⊕Z_{3}^{S}Z_{4}^{S}(4) $
R:(0,1/2,1/2)	$2R_{1}^{S}(2)⊕2R_{2}^{S}(2) $
X:(0,1/2,0)	$2X_{1}^{S}(2)⊕2X_{2}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕Λ_{2}^{S}Λ_{2}^{S}(4) $
V:(1/2,1/2,w)	$2V_{1}^{S}(2)⊕V_{2}^{S}V_{2}^{S}(4) $
D:(u,v,0)	$4D_{1}^{S}(2) $
Δ:(0,v,0)	$4Δ_{1}^{S}(2) $
E:(u,v,1/2)	$4E_{1}^{S}(2) $
S:(u,u,1/2)	$4S_{1}^{S}(2) $
Σ:(u,u,0)	$4Σ_{1}^{S}(2) $
T:(u,1/2,1/2)	$4T_{1}^{S}(2) $
U:(0,v,1/2)	$4U_{1}^{S}(2) $
W:(0,1/2,w)	$2W_{1}^{S}(2)⊕2W_{2}^{S}(2) $
Y:(u,1/2,0)	$4Y_{1}^{S}(2) $
B:(0,v,w)	$4B_{1}^{S}(2) $
C:(u,u,w)	$4C_{1}^{S}(2) $
F:(u,1/2,w)	$4F_{1}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $